1. Field of the Invention (Technical Field)
The present invention relates to phase retrieval in wave phenomena. The technical field of this invention is multidimensional phase retrieval. The preferred embodiment of this invention applies to ultrafast laser diagnostics.
2. Background Art
Interference phenomena are produced through the interaction of at least two spatially distinguishable waves. Sometimes diffraction and interference are not clearly distinguished. Interference occurs when two or more wavefronts interact while diffraction occurs naturally when a single wave is limited in some way. The nature of an interference pattern depends on several factors including the amplitudes and phases of the incoming waves. If the incoming waves are in phase, then the amplitude of the waves may add whereas, if the phase of the incoming waves are separated by 180.degree., then cancellation will result. These phenomena, the adding and canceling of amplitude, are typically referred to as constructive and destructive interference. For example, if light from a single laser is split such that it produces two separate beams, and these beams travel two different paths, then it is likely that the beams are no longer in phase or phase coherent. If the two beams are then recombined, the resulting image will be an interference pattern. In some instances, it is helpful to detect the interference pattern as an image in a plane or on a planar surface. However, the interference pattern can also be captured as intensity (absolute value of amplitude squared) versus time. The intensity versus time curve, resulting from the interference between the two beams, can be reconstructed if the individual frequency components are known i.e., amplitudes and phases.
Phase differences in the interference waves can be represented as shifts in origin. For example, the Fourier transform of a function f(x) (which converts a wave from amplitude as a function of time or space to amplitude as a function of frequency) is as follows: EQU F(k)=.intg.f(x)exp(-ikx)dx (1)
where the limits on the integral are from x=-.infin. to x=+.infin.. When the origin, or phase, is shifted, the Fourier transform is represented as, EQU F.sub.1 (k)=.intg.f(x-x.sub.o)exp-ikx)dx (2)
or as, EQU F.sub.1 (k)=F(k)exp(--ikx.sub.o) (3).
Thus, the form of the Fourier transform differs only by the phase factor exp(-ikx.sub.o), remembering that the amplitudes .vertline.F.sub.1 (k).vertline. and .vertline.F(k).vertline. and the intensity, the amplitudes squared, are equal. In many wave problems the function f(x) is complex, i.e., f(x)=Re[f(x)]+ilm[f(x)], and an analysis of transforms of complex functions applies. A plane wave has a wavefront in a plane of constant phase normal to the direction of propagation. Often a plane wave may be written as EQU E=E.sub.o exp(-i.omega.t) (4)
In Equation 4, the wave is represented as E, as a function of time, where the frequency characteristics are captured in the exponential term. In Equation 4, the amplitude is at a maximum at t=0.
Unfortunately, if phase information for the two pulses is not known ahead of time, then there is no unique combination of frequencies and amplitudes that may be combined to produce an identical plot of intensity versus time. Essentially, there is not enough information in the intensity versus time plot alone to be able to reconstruct the phase of the original pulses. The method to solve for the phase in these types of problems is referred to as phase retrieval. In essence, the phase retrieval problem is similar to solving a single equation for two unknowns--many solutions exist. To overcome this problem, a constraint must be imposed, i.e., an additional equation. Additionally, the constraint must be physically reasonable. Most phase retrieval problems are solved through imposition of a reasonable constraint that leads to a unique solution. The type of constraint depends on the application. In crystallography, symmetry conditions are typically imposed. In most instances for crystallographic application, the symmetry conditlon is applied to what is known as the outer bounds of the region. Such constraints benefit from prior knowledge of the way atoms are arranged in a crystalline structure. For other applications, other constraints must be found in order to retrieve phase.
To begin solution of such problems, an initial estimate of the phase is necessary. However, the guess is not so critical and any reasonable starting point can be used to obtain a solution. Of course, knowing that many of the solution techniques use iterative processes, a more reasonable guess typically results in fewer iterations in arriving at a unique solution.
In the field of phase retrieval, there is a class of problems known as Reconstruction from Multiple Fourier Intensities (RMI). This problem involves the reconstruction of two functions from multiple Fourier intensities of the product of relative displacements of the two functions. The solution to this important problem has significant applications to any situation where the intensity of the Fourier transform of the product of two functions is recorded for multiple relative displacements of the functions including transmission microscopy (optical, electron, and x-ray) and ultrafast laser diagnostics. For example, in the field of ultrafast laser diagnostics, the solution to the RMI problem is known as a frequency-resolved optical gatng (FROG) trace inversion, spectrogram inversion, or sonogram inversion and is used to find the intensity and phase of an ultrashort laser pulse. In the case of FROG, for example (see FIG. 1), a gate pulse which is one function is scanned, in time, across a pulse (the other function) to be measured. For each time delay, the spectrum of the pulse that results from a well-defined nonlinear interaction of these two pulses is recorded. (The nonlinear interaction produces the product of the gate and pulse.) The resulting spectrogram, or FROG trace, is a plot of intensity versus time and frequency of the pulse. Unfortunately, it is only possible to obtain intensity information of the spectrum. Consequently, the key parameters of the pulse, the intensity and phase, cannot be obtained directly from this plot. An iterative two-dimensional phase retrieval method must be used to find the phase in order to extract the functions, and hence, the pulse characteristics from its spectrogram. Methods currently exist, but they require a priori knowledge of the gate function or are slow and cumbersome, requiring large amounts of computational power to arrive at the result. There is a need for fast inversion methods for ultrashort pulse measurement devices, and the same method will be generally applicable to other fields.
Ultrafast laser systems have a large number of applications in biochemistry, chemistry, physics, and electrical engineering. These systems generate laser pulses with durations of 10 picoseconds or less and such systems are used to explore kinetics in proteins, examine carrier relaxation in semiconductors, or image through turbid media. They are also used as an ultrafast probe in electronic circuits. By using ultrafast diagnostic systems, highly advanced semiconductors, electronic circuitry, and even biomedical products can be developed and tested for commercial applications. Furthermore, new applications requiring shaped ultrashort pulses in both intensity and phase such as coherent control of chemical reactions are beginning to be developed. The continued development of these applications will require, fast, high quality, and easy-to-use ultrafast laser pulse diagnostics.
FROG is an ultrafast laser diagnostic that is used to measure the intensity and phase of an ultrashort laser pulse. In a simple form, it produces a spectrogram of the pulse that is a 3-D plot of intensity versus frequency and time delay, showing the spectral components of time slices of the pulse. While the spectrogram of the pulse serves as an intuitive display of the pulse, it is difficult to obtain quantitative information about the pulse from the spectrogram, and subtleties in the pulse structure may go unnoticed without knowledge of the actual pulse. To obtain the original pulse from its spectrogram, the phase of the spectrogram must be determined requiring a 2-D phase retrieval computation. This mathematical step--which converts the measured spectrogram into two-dimensional plots of pulse duration and chirp--is the slowest step in existing FROG instrumentation. While it is possible to characterize an individual femtosecond pulse, the data analysis step takes sixteen-orders of magnitude longer: about 1 minute on a 100 MHz Pentium-based computer. Simple adjustments become difficult. A simple error analysis may take hours. Clearly, more than just faster computers are needed to invert FROG traces in real time. An entirely new approach to method design is required.
The development of techniques for ultrashort pulse measurement, that is, the profiling of the electric field envelope and the instantaneous frequency, has proven to be difficult. Early methods yielded only the intensity autocorrelation of the pulse. Later developments, such as interferometric autocorrelation, achieved the indirect determination of various phase distortions common to ultrashort pulses, but complete intensity and phase information about the pulse remain difficult to obtain. Some work has been done to extract the time-dependent intensity I(t) and the phase .phi.(t) (or, essentially equivalent to the phase, the instantaneous frequency .omega.(t)), from these traces using iterative methods. Fundamental inherent ambiguities, including the direction of time, however, remain. It is therefore not possible to determine, for example, the sign of the chirp, unless a second measurement is made after pulse propagation through a known dispersive medium. Other methods yield only I(t) or require a streak camera and hence lack sufficient temporal resolution. Still other methods have been developed to measure the phase .omega.(t) but do not yield the intensity. Indeed, simultaneous time and frequency information is required for retrieval of the full complex electric field.
Time-frequency measurements of ultrashort pulses were first completed by Treacy in 1971. The Treacy method disperses the input pulse in frequency, selects a portion of the frequency components to produce another pulse, then cross correlates the newly formed pulse with the original input pulse. By scanning the frequency filter over all of the frequencies contained in the original pulse, a 3-dimensional plot of intensity versus frequency and time is produced which is commonly referred to as a sonogram. This method was refined by Chilla and Martinez with the development of frequency domain phase measurement or FDPM. Since the arrival time (i.e., the peak) of each frequency filtered pulse is given by the derivative of the phase (in the frequency domain) with respect to frequency, integration of arrival time of each pulse with respect to frequency gives the phase of the pulse in the frequency domain. Coupling this result with the spectrum of the pulse gives the Fourier transform of the complex electric field. The principal difficulty with this method is that if the peak of the arrival time of each frequency selected pulse does not produce a function, as is the case with self-phase modulated pulses, the group delay is not well defined and characterization of the pulse is not possible.
Spectrograms are close relatives to sonograms. Rather than displaying the time arrival of frequency filtered pulses, a spectrogram displays the frequency content of time slices of a pulse. Using optical methods to obtain a spectrogram of the pulse to be measured is the basis of a relatively new technique for the complete characterization of ultrashort laser pulses called frequency-resolved optical gating (FROG). FROG is a versatile technique that can be used in either multi-shot or single shot geometries. A gate pulse, which can be virtually any duration, slices out portions of a probe pulse in the time domain using either an instantaneously responding nonlinear material or a nearly instantaneously responding medium. The sampled portion of the probe, or signal, is dispersed in a spectrometer. Like a sonogram, the resulting spectrogram contains all the intensity and phase information about the probe pulse.
Obtaining the spectrogram of a pulse is experimentally less complex than obtaining the sonogram; however, extracting the intensity and phase of a pulse from its spectrogram is mathematically more challenging. If an approach similar to that of Chilla and Martinez is used to invert a spectrogram, the phase in the time domain is obtained, but the complete complex electric field is not. Since only the magnitude of the spectrogram can be measured, finding the full intensity and phase of the input pulse requires determining the spectrogram's phase, placing the FROG inversion problem into the category of 2-dimensional phase retrieval problems.
Frequency-resolved optical gating (FROG) is a technique used to measure the intensity and phase of an ultrashort laser pulse without ambiguity; it is broadband and does not require phase matching. Whereas Chilla and Martinez measured the cross correlation of a particular frequency component of an ultrashort pulse, FROG involves measuring the spectrum of a particular temporal component of the pulse (see FIG. 1). FROG does this by spectrally resolving the signal pulse in virtually any autocorrelation-type experiment performed in an instantaneously responding nonlinear medium.
As shown in FIG. 1, FROG involves splitting a pulse and overlapping the two resulting pulses in an instantaneously responding X.sup.(3) (or X.sup.(2) may be used as well, however, in this case, information about the direction of time is lost). Consequently, when the spectrogram is inverted and the pulse characteristics extracted, there is an ambiguity in direction of time. Even though any instantaneous nonlinear interaction may be used to implement FROG, perhaps the most intuitive is the polarization-gating configuration. In this case, the induced birefringence due to the electronic Kerr effect is used as the nonlinear-optical process. In other words, the "gate" pulse causes the X.sup.(3) medium, which is placed between two crossed polarizers, to become slightly birefringent. The polarization of the "gated" pulse (which is cleaned up by the first polarizer) is rotated slightly by the induced birefringence allowing some of the "gated" pulse to leak through the second polarizer. This is referred to as the signal. Because most of the signal emanates from the region of temporal overlap between the two pulses, the signal pulse indicates the frequencies of the gated pulse within this overlap region (See FIG. 1 inset). The signal is then spectrally resolved, and the signal intensity is measured as a function of wavelength and delay time T. The resulting trace of intensity versus delay and frequency is a spectrogram, a time- and frequency-resolved transform that intuitively displays time-dependent spectral information of a waveform.
The spectrogram can be expressed as: ##EQU1##
where E(t) is the measured pulse's electric field, g(t-.tau.) is the variable-delay gate pulse, and the subscript E on S.sub.E indicates the spectrogram's dependence on E(t). The gate pulse g(t) is usually somewhat shorter in length than the pulse to be measured, but not infinitely short. This is an important point: an infinitely short gate pulse yields only the intensity I(t) and conversely, a continuous wave ("CW") gate yields only the spectrum I(.omega.). On the other hand, a finite-length gate pulse yields the spectrum of all of the finite pulse segments with duration equal to that of the gate. While the phase information remains lacking in each of these short-time spectra, this loss is compensated by having the spectrum of an infinitely large set of pulse segments. The spectrogram has been shown to nearly uniquely determine both the intensity I(t) and phase .phi.(t) of the pulse, even if the gate pulse is longer than the pulse to be measured (although if the gate is too long, sensitivity to noise and other practical problems arise).
In FROG, when using optically induced birefringence as the nonlinear effect, the signal pulse is given by: EQU E.sub.sig (t, .tau.).varies.E(t).vertline.E(t-.tau.).vertline..sup.2 (6)
so the measured signal intensity I.sub.FROG (.omega., .tau.), after the spectrometer is: ##EQU2##
We see that the FROG trace is thus a spectrogram of the pulse E(t) although the gate pulse .vertline.E(t).vertline..sup.2 is a function of the pulse itself. For Second Harmonic Generation ("SHG") FROG, E(t), the pulse itself, rather than .vertline.E(t).vertline..sup.2, is the gate function.
To see that the FROG trace essentially uniquely determines E(t) for an arbitrary pulse, it is first necessary to observe that E(t) is easily obtained from E.sub.sig (t, .tau.). Then it is simply necessary to write equation (7) in terms of E.sub.sig (t, .OMEGA.), the Fourier transform of the signal field E.sub.sig (t, .tau.) with respect to delay variable T. We then have what appears to be a more complex expression, but one that will give us better insight into the problem: ##EQU3##
Equation (8) indicates that the problem of inverting the FROG trace I.sub.FROG (.omega., .tau.) to find the desired quantity E.sub.sig (t, .OMEGA.) is that of inverting the squared magnitude of the two-dimensional (2-D) Fourier transform of E.sub.sig (t, .OMEGA.). This problem, which is called the 2-D phase-retrieval problem, is well known in many fields, especially in astronomy, where the squared magnitude of the Fourier transform of a 2-D image is often measured. At first glance, this problem appears unsolvable; after all, much information is lost when the magnitude is taken. Worse, it is well known that the 1-D phase retrieval problem is unsolvable (for example, infinitely many pulse fields give rise to the same spectrum). Intuition fails badly in this case, however; two- and higher dimensional phase retrieval essentially always yields unique results. At this point, it should be noted that FROG data are usually collected using a CCD so that the integrals shown in Eqs. 6, 7 and 8 are easily replaced by sums.
Since the FROG trace inversion problem is a 2-D phase retrieval problem, an iterative method is required to find the phase. Ideally, each iteration results in a slightly better "guess" for the solution of the phase until convergence. How a method determines each subsequent guess is paramount to it performance. The first FROG inversion method used integration of E(t) .vertline.E(t-.tau.).vertline..sup.2 with respect to .tau. to obtain subsequent guesses for E(t). While fast, this method stagnates easily and fails to invert spectrograms of double pulses.
To overcome stagnation problems in the FROG inversion DeLong et al. developed an improved composite method that employs modifying constraints in the basic method and a multivariate minimization of the FROG trace error with respect to E(t): ##EQU4##
where .epsilon..sub.FT is the per element RMS error of the FROG trace, I.sub.FT (.omega..sub.i, .tau..sub.j) is the current iteration of the FROG trace (constructed from the current E(t)), I.sub.MEASURED (.omega..sub.i, .tau..sub.j) is the measured FROG trace, and .omega..sub.i and .tau..sub.j are the i.sup.th frequency and jt delay in the frequency and delay vectors, respectively. Three separate minimization methods are used to help prevent stagnation of the compound method. Unfortunately, multivariate minimization methods are very slow, sometimes requiring almost an hour to converge.
The method of generalized projections provided a much-needed boost to the speed of the compound FROG inversion method, obviating the need for multvariate minimization. Generalized projections is a powerful technique that works to solve systems where the solution lies at the intersection of two a more sets. In the case of FROG, set one is the set of all complex spectrograms with the same magnitude as the spectrogram to be inverted. The other set, set two, is the set of all pulses that fits the physics of the construction of the optical spectrogram. The fastest FROG inversion methods iterate between members of these two sets, but in the case of GP, a member of one set is chosen that minimizes the distance between it and the member of the other set. This is accomplished by producing a new guess for E(t) that minimizes the error function: ##EQU5##
with respect to E.sub.i (t) where i is the iteration number.
The complete GP method present by DeLong et al. works as follows: An initial guess of random noise modulated by a Gaussian is used for E(t) to generate a spectrogram, E.sub.sig (.omega., .tau.). During each iteration of the method, the magnitude of E.sub.sig (.omega., .tau.) is replaced by the square root of the experimentally measured FROG trace. To find the next guess for E(t) an inverse Fourier transform is performed to obtain E'.sub.sig (t, .tau.). The next guess for E(t) is determined by the GP step described above. The method is repeated until the error reaches an acceptable minimum.
The GP method presented by DeLong et al. is robust, inverting almost any spectrogram while being faster than brute force minimization. However, for simple pulses, it is slower than the basic FROG method because it still contains a minimization step (albeit along the gradient of Z). Consequently, the complete FROG inversion method presented by DeLong et al. is a composite method that still uses all the basic FROG method because the basic method is faster than the GP method for some pulses.
References that appear to disclose phase retrieval techniques are as follows: Method and Apparatus for Measuring the Intensity and Phase of an Ultrashort Light Pulse, U.S. Pat. No. 5,754,292, Kane et al. (May 19, 1998). This patent discloses an iterative process for determining time-dependent intensity and phase of a spectrogram obtained from frequency-resolved optical grating (FROG) of an ultrashort light pulse. The FROG technique yields an "experimental" intensity signal dependent on frequency and the delay time between a gate pulse and a probe pulse. Phase information is obtained from the two-dimensional intensity data through a phase-retrieval method. The patent discloses a preferred iterative one dimensional Fourier transform method. The preferred method performs a Fourier transform from time domain to frequency domain and an inverse Fourier transform from frequency domain to time domain. The method also requires an initial guess for the time dependent electrical field. According to the disclosure, an electrical field based on noise suffices for the initial guess (although an alternative method for deriving an initial guess is discussed). This guess is then used to create a time and delay time dependent electrical field. A one-dimensional Fourier transform of this field yields a frequency and delay time dependent electrical field. The magnitude of this field is replaced with the square root of the "experimental" frequency and time delay dependent intensity signal. An inverse Fourier transform is performed to yield a time and delay time dependant electrical field. This field is integrated with respect to delay time to update the initial guess. The iterative process continues until convergence. The disclosure does not state the convergence criteria.
Method and Apparatus for Measuring the Intensity and Phase of One or More Ultrashort Light Pulses and for Measuring Optical Properties of Materials, U.S. Pat. No. 5,530,544, Trebino et al. (Jun. 25, 1996). In a first embodiment, for a single pulse, this patent discloses a phase-retrieval solution method similar to that of the '292 Kane et al. patent. However, an initial guess using a Guassian-intensity flat-phase pulse is said to provide better convergence. The disclosure states that the difference between the experimental FROG trace and the FROG trace generated by the calculated electric field serves as a meaningful measure of error for purpose of a convergence criterion. This is quantified in the method as a RMS error value. In a second embodiment, for multiple pulses, this patent states that phase-retrieval has proven to be "extremely difficult" for the basic FROG method. To overcome basic FROG limitations, the method adds an intensity constraint, an overcorrection method, and a multidimensional minimization technique. In a third embodiment, this patent discloses a method for phase-retrieval based on the theory of Generalized Projections (GP). The GP method relies on satisfying two distinct mathematical constraints. The correct signal field lies at the intersection of the two constraint sets. The intersection of the two constraint sets is found by iteratively projecting, or mapping, points onto the two constraint sets. This embodiment uses an iterative error minimization technique based on a distance function to approximately find where the points converge near the intersection of the two sets. In a fourth embodiment, the patent discloses a technique that compensates for non-instantaneous components of the response. Non-instantaneous components result from Raman absorbance and emission and may impart cubic phase distortions in the retrieved pulse frequency domain. The compensatory technique accounts for the non-instantaneous response of the medium. The added degree of accuracy increases the number of calculations required for solution. Where N.sup.2 is the number of pixels in a FROG trace, the number of calculations required by the fourth embodiment scale by N.sup.3 in comparison to N.sup.2 for the nearly instantaneous case. A fifth embodiment deviates from the prior embodiments in that it may resolve intensity and phase of more than one pulse on a multiple or single-shot basis. The fifth embodiment requires modification of the basic apparatus to accommodate non-identical gate and probe pulses. Retrieval of intensity and phase of both pulses is referred to as Twin Recovery of Electric field Envelopes using FROG, or TREEFROG. The TREEFROG problem is similar to blind deconvolution, i.e., to retrieve both the original image and distortion function from a blurred image. The process resolves the probe and gate pulses as a function of time given the 2-dimensional TREEFROG intensity plot. The TREEFROG method begins with guess for both probe and gate pulses to generate an electric field from a Fourier transform of P(t)G(t-.tau.). The magnitude squared of the electric field forms a trial TREEFROG trace. This trial trace is compared with the experimental trace to determine convergence. The magnitude of the trial is constrained to the intensity of the experimental trace while leaving the phase unchanged thus giving a modified electric field. An inverse Fourier transform of the modified signal yields a modified electric field with respect to time and delay time (a typographical error appears in the patent). The method of generalized projections is used to generate alternate updates of probe and gate pulses as a function of time by minimizing an error function with respect to the pulse of interest. A spectral constraint is added to increase the robustness of the method of the fifth embodiment. The spectral constraint requires measurement of field spectra. The sixth embodiment of the invention discloses a method using two FROG apparatuses or a TREEFROG apparatus to measure optical properties of a medium. The analysis uses a method similar to the GP TREEFROG of the fifth embodiment. In a seventh embodiment, the method disclosed in the seventh embodiment is analogous to optical heterodyne detection methods. This embodiment obviates the need for phase retrieval since the method measures the complex field and solves for both imaginary and real parts of the field. The complex field can then be used to calculate intensity and phase. An interferometric second-harmonic generation (ISHG) variation is also disclosed--both techniques use local oscillator-like pulse mixing and require minimization methods. The ISHG solution is identical to the normal FROG phase retrieval problem. The eighth embodiment involves measuring the spectrum of the coherent sum of known and unknown pulses. The method is known as Temporal Analysis of a Dispersed Pair Light E-fields (TADPOLE). Solution of two simultaneous equations with two unknowns yields phase information. A ninth embodiment discloses a solution technique based on artificial neural nets (ANN) for retrieval of pulse information from an experimental FROG trace. Training the ANN is the time consuming step. Theoretically, a trained ANN may provide pulse information on a nearly real-time basis. A variety of training enhancements is also disclosed together with disadvantages of the ANN approach.
Apparatus for Characterizing Short Optical Pulses, U.S. Pat. No. 5,684,586, Fortenberry et al. (Nov. 4, 1997). This patent discloses an analyzer for analyzing intensity and phase characteristics of an optically dispersed short input pulse. The intensity and phase characteristics of the short input pulse are determined by applying a back-propagation method to the dispersed pulse. In a preferred embodiment, the dispersed pulse is split to create an optical interference spectrogram. Analysis of the split dispersed pulse optical interference spectrogram yields intensity and phase of the dispersed pulse. The back-propagation technique relies on knowledge of the transfer function of the optical disperser. The specification states that known methods of analyzing spectrograms, such as those created by the split dispersed pulse, include the FROG technique by Kane et al. and optical heterodyne detection techniques.
Method and Apparatus for Measuring Ultrashort Optical Pulses, U.S. Pat. No. 4,792,230, Naganuma et al. (Dec. 20, 1988). In one embodiment, the method and apparatus of Naganuma et al. measures intensity shape and pulse shape of ultrashort optical pulses that repeat with a constant period. These pulses are subject to a beam splitter, two different path lengths, recombination and focusing onto a doubling crystal. The doubling crystal generates second harmonic light, i.e., light with a wavelength one half that of the incident light, that is converted to a voltage by an optical detector. Three spectra are recorded and analyzed through Fourier transform techniques. Phase reconstruction proceeds through an iterative calculation. In a second embodiment, ultrashort optical pulses repeating with a constant period are subject to a similar analysis, however, the signal recorded by the optical detector is divided into two portions. Two components are extracted from the signal, a low frequency DC component and a high frequency component near the frequency of the interference fringe near the fundamental wave--obtained through use of an AC voltmeter. The DC and AC measurements occur simultaneously while constantly varying the difference between the pulse path lengths. Fourier analysis of the data yields the intensity shape of the measured pulse while iterative calculations yield the phase shape. In a third embodiment, a detector is used that produces an electrical signal proportional to the fundamental light and the second-harmonic light. This signal is collected while constantly varying the pulse path lengths. A Fourier analysis of the intensity change data of the first harmonic and second harmonic light together with the intensity of the measured optical pulse yields intensity shape and phase shape through an iterative process. In essence, the patent discloses use of a Michelson interferometer and a doubling crystal to produce signals amenable to various autocorrelation techniques for second-harmonic electric fields.
Spatial Wavefront Evaluation by Intensity Relationships, U.S. Pat. No. 5,367,375, Siebert (Nov. 22, 1994). This patent discloses a method for determining the phase profile of a wavefront at a first plane using additional information about the wavefront at a second plane. The method entails determining impinging a wave on a first plane, measuring the intensity of the wavefront at a second plane, and determining the phase difference of the wavefront at the first plane in accordance with a transfer function that employs data gathered at the two planes. Three noniterative approaches are given to solve an optical transfer function equation for aperture phase and pupil wavefront: polynomial approach, sampling approach, and general approach.
Signal Processing Apparatus and Method for Iteratively Determining Arithmetic Fourier Transform, U.S. Pat. No. 5,253,192, Tufts (Oct. 12, 1993). This patent discloses a method and apparatus for iteratively determining the inverse Arithmetic Fourier Transform (AFT). In general, analog to digital conversion of input data limits the standard forward AFT as opposed to accumulation of rounding and coefficient errors that limit Fast Fourier Transform techniques. In addition, the AFT does not require storage of memory addressing. The AFT method is also closely related to the least squares successive approximation realization of the Discrete Fourier Transform (DFT). In one embodiment, an input signal is received and a data signal vector generated. A frequency domain signal vector is generated by multiplying the data signal vector by a predetermined transform signal matrix and a predetermined number of iterations. An inverse transformation on the frequency domain signal vector is performed by multiplying the frequency domain signal vector by an AFT signal matrix; this step produces a synthesized data signal vector. The next step generates and error signal vector that is converted into a frequency domain improvement signal by multiplying the error signal vector by the transpose of the AFT signal matrix and a predetermined scaling signal value. Finally, an updated frequency domain signal vector is generated and the iterative cycle continues until the error reaches a specified minimum. In another embodiment, an input signal is received and a data signal vector generated. A frequency domain signal vector is generated by multiplying the data signal vector by a predetermined transform signal vector rather than a predetermined transform signal matrix.